Question

The number of $ 5 - $ letter word that can be formed by using the letters of the word SARANAM is
(a) $ 1120 $
(b) $ 6720 $
(c) $ 480 $
(d) $ 720 $

Answer 1

Hint: As we know that the above question is of permutations and combinations. In this question we have to find the number of ways of forming $ 5 $ letter word of the word SARANAM. We will make different possible scenarios from the letters of the word and the number of ways we can arrange it by applying the concepts of permutations and combinations.

Complete step by step solution:
We can see that in the word SARANAM there are total $ 7 $ alphabets i.e. $ n = 7 $ .
But as we can see that there are $ 3A $ , and rest all the letters are different. So we have three different cases which are all five letters are different, $ 3 $ letters are different and $ 2 $ are alike and the third case is $ 2 $ are different and $ 3 $ are alike. So we will solve all the cases.
We have total standard $ 5 $ letters i.e. $ 3A $ area considered as one and $ S,R,N,M $ . So total ways of arranging these five letters are $ 5! $ .
In the second case we have two are alike and there are different i.e.
 $ \Rightarrow ^4{C_3} \times \dfrac{{5!}}{{2!}} $ , Since we have to choose $ r $ out of $ n $ .
In the third case $ 2 $ are different and $ 3 $ area like. So by applying the same concept as second we can write that the number of arranging the ways are
 $ \Rightarrow ^4{C_2} \times \dfrac{{5!}}{{3!}} $ .
We will add all the three ways together i.e.
 $ \Rightarrow 5!{ + ^4}{C_3}\dfrac{{5!}}{{2!}}{ + ^4}{C_2}\dfrac{{5!}}{{3!}} $ .
On further simplifying we have $ 120 + 240 + 120 = 480 $ .
Hence the correct option is (c) $ 480 $ .
Hence the total number of possible permutations in the word MISSISSIPPI are $ 34650 $ .
So, the correct answer is “Option C”.

Note: Whenever we face such a type of question we should figure all the possible methods which can arise from the letters in the formation of the word. We should know that permutations means arrangement which is simply the grouping of objects according to the requirement. We should be careful while calculating the number of ways in the word since there are few letters that are repeated and they need to be calculated together. Whenever we face such types of problems the value point to remember is that we need to have a good knowledge of permutations and its formulas.